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# PG(10)05 - Problems in Geometry(5 1 Given a circle consider...

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Problems in Geometry (5) 1. Given a circle γ, consider points P, C, D γ and let [ A, B ] be a diameter of γ . Prove that the Simson line of P with respect to the triangle ACD is perpendicular to the Simson line of P with respect to the triangle BCD. 1 2. Given a triangle ABC with circumcenter O, let A 0 , B 0 , C 0 be the respective midpoints of the line segments [ B, C ] , [ C, A ] , [ A, B ] . Let E be the foot of the perpendicular from B on CA and U, W be the feet of perpendiculars from E on B 0 C 0 , A 0 B 0 respectively. Prove that WU bisects [ B, E ] and is 2 parallel to OB . 3. Given a triangle ABC, let the parallel to BC through A meet the circumcircle of ABC in L for a second time. (A) Prove that the Simson line of L is parallel to OA where O is the circumcenter of ABC. (B) Let [ A, V ] and [ L, U ] be diameters of the circumcircle of ABC. Describe the Simson lines of U and V. 4. Consider a triangle ABC with circumcenter O . For each X on the circumcircle of ABC let l X denote the Simson line of X with respect to ABC. (A) Prove that in order for l X
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